Let $M_k(N)$ be the space of modular forms of level $N$ and weight $k$ whose $q$-coefficients at infinity are rational and $l$-integral.
Put $\tilde M_k(N) = M_k(N) \otimes \mathbb{\bar F}_l$ and let $\mathbb T_k$ be the subring of $\mathrm{End}_{\mathbb{\bar F}_l}(\tilde M_k(N))$ generated by Hecke operators $T_p$ for all prime $p$, prime to $N$.
What is the number of different systems of eigenvalues for all $T_p$ with $(p,N)=1$?
My feeling is that this is a number of maximal ideals in $\mathbb T_k$, but can not make it precise.

1 Answer
1

Your intuition is correct. Actually this has nothing to do with modular forms: there is a simple general statement which contains your question:

Let $k$ be an algebraically closed field,
$M$ a finite dimensional $k$-vector space, $(T_i)$ a family of commuting
linear operators on $M$, and $\tau$ the sub-algebra of $End_R(M)$ they generate.
Then there is a natural bijection between the maximal ideals of $\tau$ and the system of eigenvalues for all the $T_i$ in $M$.

The bijection is easily constructed in one direction: suppose that $T_i \mapsto \lambda_i$
is a system of eigenvalues appearing in $M$ (that is to say, there is a $0 \neq v \in M$ such that $T_i v = \lambda_i v$ for all $i$). Then obviously, $v$ is an eigenvector
for all $T \in \tau $ and the map $ \tau \rightarrow k$ sending $T \in \tau$ on the unique
$\lambda(T) \in k$ such that $T v = \lambda(T) v$ is a morphism of $k$-algebras.
Its kernel is thus a maximal ideal of $T$. This defines a map "system of eigenvalues" $\rightarrow$ "maximal ideals of $\tau$".

Conversely, if $m$ is a maximal ideal of $\tau$, then $\tau/m=k$ (since $\tau$ is finitely generated and $k$ is algebraically closed) so the map $\lambda : \tau \rightarrow \tau/m=k$ is a morphism of $k$-algebras. You want to prove that there is a $0 \neq v \in M$ such that $T v = \lambda(T)v$ for all $T \in \tau$. It suffices to do it for a family of generators
$T_1,\dots,T_k$ of $\tau$, which we can choose finite. Now since $\lambda (T_1-\lambda(T_1))=0$, $T_1-\lambda(T_1)$ is not invertible in $\tau$, hence is not invertible
in $End_k(M)$ (otherwise its inverse would be by Cayley-Hamilton a polynomial in $T_1$,
hence in $\tau$) hence $V_1 = \ker (T_1 - \lambda(T_1))$ is not $0$. Now $T_2, \dots, T_k$
stabilize $V_1$ since they commute with $T_1$, and you can make the same reasoning for $T_2$ acting on $V_1$, and so on. At the end you get a non-zero subspace of $V$ on which
each $T_i$ acts by the scalar $\lambda(T_i)$, QED.

That's a very pedestrian proof. If you are willing to apply some basic commutative algebra
(e.g. Artinian rings are semi-local) you can do it much faster.